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<p>This paper investigates a class of neutral stochastic McKean–Vlasov functional differential equations with mixed multiple impulse effects. Compared with classical models, the studied equation demonstrates significant theoretical innovations: its drift term and diffusion term are not only dependent on time-varying parameters and the current system state but also exhibit coupling relationships with the probability distribution of the system state. By constructing a comprehensive theoretical analysis framework, this paper achieves the following innovative results: Regarding the existence and uniqueness of solutions, the discontinuity problem of the solution trajectory is decomposed using impulse interval segmentation technology. Under the global Lipschitz continuity assumptions, stochastic analysis tools including <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11420_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\ddot{o}lder's\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mover accent="true"> <mi>o</mi> <mo>¨</mo> </mover> <mi>l</mi> <mi>d</mi> <mi>e</mi> <msup> <mi>r</mi> <mo>′</mo> </msup> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation> inequality and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11420_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(It\hat{o}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mi>t</mi> <mover accent="true"> <mi>o</mi> <mo stretchy="false">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> isometry are comprehensively applied, combined with the Banach fixed-point theorem to establish well-posedness criteria for such complex systems. For stability analysis, this work innovatively incorporates mixed multiple impulses. Through quantitative methods based on impulse density functions and Gronwall inequality derivations, criteria for mean-square asymptotic stability of the system are rigorously established. These results are further extended to mean-square exponential stability, explicitly revealing quantitative relationships among impulse intensity, impulse density, and system stability. Finally, two numerical examples are provided to verify the effectiveness of the theoretical results.</p>

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Well-posedness and stability of neutral stochastic McKean–Vlasov functional differential equation with multiple impulses

  • 期刊论文

  • 发表日期：

    2025年6月7日

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  • Junwen Wan,
  • Quanxin Zhu

摘要

This paper investigates a class of neutral stochastic McKean–Vlasov functional differential equations with mixed multiple impulse effects. Compared with classical models, the studied equation demonstrates significant theoretical innovations: its drift term and diffusion term are not only dependent on time-varying parameters and the current system state but also exhibit coupling relationships with the probability distribution of the system state. By constructing a comprehensive theoretical analysis framework, this paper achieves the following innovative results: Regarding the existence and uniqueness of solutions, the discontinuity problem of the solution trajectory is decomposed using impulse interval segmentation technology. Under the global Lipschitz continuity assumptions, stochastic analysis tools including \(H\ddot{o}lder's\) H o ¨ l d e r ′ s inequality and \(It\hat{o}\) I t o ^ isometry are comprehensively applied, combined with the Banach fixed-point theorem to establish well-posedness criteria for such complex systems. For stability analysis, this work innovatively incorporates mixed multiple impulses. Through quantitative methods based on impulse density functions and Gronwall inequality derivations, criteria for mean-square asymptotic stability of the system are rigorously established. These results are further extended to mean-square exponential stability, explicitly revealing quantitative relationships among impulse intensity, impulse density, and system stability. Finally, two numerical examples are provided to verify the effectiveness of the theoretical results.